Optimal Control of Nonholonomic Systems via Magnetic Fields

TL;DR

This paper develops a geometric optimal control framework for nonholonomic mechanical systems influenced by magnetic fields, deriving necessary conditions for optimality and analyzing a magnetically controlled Chaplygin Sleigh.

Contribution

It introduces a novel control approach that manipulates the magnetic field via the symplectic form, extending geometric control methods to magnetic control of nonholonomic systems.

Findings

Derived necessary optimality conditions for magnetic control systems.

The magnetically controlled Chaplygin Sleigh problem is shown to be completely integrable.

Provides a foundation for future magnetic control applications in nonholonomic systems.

Abstract

Geometric optimal control utilizes tools from differential geometry to analyze the structure of a problem to determine the control and state trajectories to reach a desired outcome while minimizing some cost function. For a controlled mechanical system, the control usually manifests as an external force which, if conservative, can be added to the Hamiltonian. In this work, we focus on mechanical systems with controls added to the symplectic form rather than the Hamiltonian. In practice, this translates to controlling the magnetic field for an electrically charged system. We develop a basic theory deriving necessary conditions for optimality of such a system subjected to nonholonomic constraints. We consider the representative example of a magnetically charged Chaplygin Sleigh, whose resulting optimal control problem is completely integrable.